Lines & Planes in Space · Hard problem

A searchlight beam travels in a straight line along the direction in which the planes $2x+y-z=3$ and $x-y+2z=1$ meet, starting from the point on that line where $z=0$ and heading in the direction of increasing $x$. After the beam has covered a distance of $3\sqrt{35}$ units, find its $z$-coordinate.